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Properties

Label 1-4024-4024.451-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.656 + 0.754i$
Analytic cond. $18.6873$
Root an. cond. $18.6873$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.877 + 0.479i)3-s + (0.947 + 0.319i)5-s + (0.958 − 0.283i)7-s + (0.539 + 0.842i)9-s + (0.452 + 0.891i)11-s + (0.883 − 0.468i)13-s + (0.677 + 0.735i)15-s + (−0.640 + 0.768i)17-s + (0.155 − 0.987i)19-s + (0.977 + 0.211i)21-s + (0.934 − 0.355i)23-s + (0.795 + 0.605i)25-s + (0.0687 + 0.997i)27-s + (−0.570 − 0.821i)29-s + (−0.787 − 0.615i)31-s + ⋯
L(s)  = 1  + (0.877 + 0.479i)3-s + (0.947 + 0.319i)5-s + (0.958 − 0.283i)7-s + (0.539 + 0.842i)9-s + (0.452 + 0.891i)11-s + (0.883 − 0.468i)13-s + (0.677 + 0.735i)15-s + (−0.640 + 0.768i)17-s + (0.155 − 0.987i)19-s + (0.977 + 0.211i)21-s + (0.934 − 0.355i)23-s + (0.795 + 0.605i)25-s + (0.0687 + 0.997i)27-s + (−0.570 − 0.821i)29-s + (−0.787 − 0.615i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.656 + 0.754i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.656 + 0.754i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.590037749 + 1.636172963i\)
\(L(\frac12)\) \(\approx\) \(3.590037749 + 1.636172963i\)
\(L(1)\) \(\approx\) \(1.970937427 + 0.4972817999i\)
\(L(1)\) \(\approx\) \(1.970937427 + 0.4972817999i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.877 + 0.479i)T \)
5 \( 1 + (0.947 + 0.319i)T \)
7 \( 1 + (0.958 - 0.283i)T \)
11 \( 1 + (0.452 + 0.891i)T \)
13 \( 1 + (0.883 - 0.468i)T \)
17 \( 1 + (-0.640 + 0.768i)T \)
19 \( 1 + (0.155 - 0.987i)T \)
23 \( 1 + (0.934 - 0.355i)T \)
29 \( 1 + (-0.570 - 0.821i)T \)
31 \( 1 + (-0.787 - 0.615i)T \)
37 \( 1 + (0.217 + 0.976i)T \)
41 \( 1 + (-0.474 - 0.880i)T \)
43 \( 1 + (-0.871 + 0.490i)T \)
47 \( 1 + (0.0562 + 0.998i)T \)
53 \( 1 + (-0.155 - 0.987i)T \)
59 \( 1 + (0.192 + 0.981i)T \)
61 \( 1 + (0.0312 + 0.999i)T \)
67 \( 1 + (0.677 - 0.735i)T \)
71 \( 1 + (0.852 + 0.523i)T \)
73 \( 1 + (0.910 - 0.412i)T \)
79 \( 1 + (-0.764 - 0.644i)T \)
83 \( 1 + (-0.998 - 0.0625i)T \)
89 \( 1 + (-0.920 - 0.389i)T \)
97 \( 1 + (-0.905 - 0.424i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.49940453351806123070181474667, −17.94102294995306575674094519797, −17.028251740831939601569784997055, −16.44165495216926113349277822200, −15.58074216948515008892965526953, −14.73878534486539053129059844244, −14.02227928873809665575513241942, −13.90318649025542178980520714223, −13.01987529874275124873060043267, −12.41530921769733524257444884750, −11.39247691128711467020534462853, −10.99256041653979289746385247538, −9.85365312788642369793182769583, −9.089397447102956474483113974960, −8.73600572508404764715333658175, −8.14954966473757952815637991336, −7.11969536645220882289218777279, −6.52377991173804970191096336466, −5.62741992190124686092677769577, −5.02970822243792151411671861644, −3.915786667025858019739966555630, −3.25754727162912787527109474933, −2.248935401275188418737647428531, −1.58463371062908159735369504265, −1.06830171425523753839872016608, 1.25490646281247701948697175063, 1.89967232850026457915025223479, 2.57265915242955977567922256048, 3.5145674561852119255886925948, 4.34420357209892001450311277571, 4.921811883112746342072935369733, 5.79352675253138894331582033629, 6.77955983557847687655280870491, 7.355695196379280300099616230268, 8.32279152061543506949731331892, 8.81823853440435631995913296931, 9.58248161814821053157256892983, 10.179304199091759522338350846468, 10.989940097165725059345540033734, 11.32172686611499789754872262244, 12.753617056200203554936329442032, 13.27177154229324275645501747553, 13.79308241437299179331945759944, 14.59112763379247563509471567101, 15.128043950958622963939014206471, 15.44367574862863859774101980523, 16.74106096079877019174448789922, 17.18365208958754083269411862450, 17.924253604359847278526229765511, 18.43307258553240039036908522783

Graph of the $Z$-function along the critical line